{ "cells": [ { "cell_type": "code", "execution_count": null, "id": "306f3472", "metadata": {}, "outputs": [], "source": [ "%matplotlib inline\n", "import time\n", "import numpy as np\n", "import matplotlib as mpl\n", "import matplotlib.pyplot as plt\n", "import seaborn as sns\n", "\n", "sns.set_style(\"whitegrid\")\n", "sns.set_palette(\"colorblind\")\n", "palette = sns.color_palette()\n", "figsize = (10, 6)\n", "legend_fontsize = 13\n", "\n", "from matplotlib import rc\n", "rc('font', **{'family': 'sans-serif'})\n", "rc('figure', **{'dpi': 120})\n", "\n", "rng = np.random.default_rng(4)" ] }, { "cell_type": "markdown", "id": "4c677c9e", "metadata": {}, "source": [ "# Как обучаются нейросети" ] }, { "cell_type": "markdown", "id": "a10c55fb", "metadata": {}, "source": [ "## 1. Где мы остановились: производная без матана" ] }, { "cell_type": "code", "execution_count": null, "id": "56c5ecc3", "metadata": {}, "outputs": [], "source": [ "def slope(f, c, eps=1e-4):\n", " \"\"\"Численная производная: чуть-чуть вправо, чуть-чуть влево, поделить на разницу.\"\"\"\n", " return (f(c + eps) - f(c - eps)) / (2 * eps)\n", "\n", "\n", "E = lambda c: (c - 30) ** 2\n", "\n", "print(\" c по формуле 2(c−30) «чуть-чуть повернуть»\")\n", "for c in [10, 20, 25, 30, 35, 50]:\n", " print(\" %2d %18.3f %20.3f\" % (c, 2 * (c - 30), slope(E, c)))" ] }, { "cell_type": "code", "execution_count": null, "id": "9beeaacf", "metadata": {}, "outputs": [], "source": [ "def gradient_descent(f, c0, eta, n_steps):\n", " cs = [c0]\n", " for _ in range(n_steps):\n", " cs.append(cs[-1] - eta * slope(f, cs[-1]))\n", " return np.array(cs)\n", "\n", "\n", "trace = gradient_descent(E, 20., eta=0.1, n_steps=10)\n", "cs = np.linspace(10, 50, 200)\n", "fig, ax = plt.subplots(figsize=figsize)\n", "ax.plot(cs, E(cs), linewidth=2.5, color=\"0.4\", label=\"E(c) = (c − 30)²\")\n", "ax.plot(trace, E(trace), \"o-\", color=palette[1], linewidth=2, markersize=8,\n", " label=\"шаги c ← c − 0,1 · (чуть-чуть повернуть)\")\n", "ax.set_xlabel(\"предполагаемая цена страницы c\"); ax.set_ylabel(\"квадрат промаха\")\n", "ax.set_title(\"Тот же спуск, что в прошлый раз, — но производная посчитана без формулы\")\n", "ax.legend(fontsize=11)\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": null, "id": "014a2cc1", "metadata": {}, "outputs": [], "source": [ "pages = np.array([1., 2., 3., 4., 5., 6., 7., 8., 8., 9., 10., 11., 12., 12.])\n", "color_pages = rng.integers(0, 3, len(pages)) # этого в чеке нет\n", "paid = 20 + 30 * pages + 15 * color_pages\n", "\n", "\n", "def predict2(w, x):\n", " \"\"\"Правило с двумя настройками: плата за заказ b = w[0] и цена страницы c = w[1].\"\"\"\n", " return w[0] + w[1] * x\n", "\n", "\n", "def mean_sq2(w, x=pages, y=paid):\n", " return np.mean((y - predict2(w, x)) ** 2)\n", "\n", "\n", "X = np.vstack([np.ones_like(pages), pages]).T\n", "w_star = np.linalg.solve(X.T @ X, X.T @ paid) # дно, для проверки\n", "print(\"Дно ландшафта по формуле: плата за заказ %.2f ₽, страница %.2f ₽, ошибка %.1f\"\n", " % (w_star[0], w_star[1], mean_sq2(w_star)))" ] }, { "cell_type": "markdown", "id": "67aab900", "metadata": {}, "source": [ "## 2. Обещание первое: почему квадрат, а не модуль" ] }, { "cell_type": "code", "execution_count": null, "id": "fcc3af72", "metadata": {}, "outputs": [], "source": [ "def mean_abs_c(c):\n", " return np.mean(np.abs(paid - c * pages))\n", "\n", "\n", "def mean_sq_c(c):\n", " return np.mean((paid - c * pages) ** 2)\n", "\n", "\n", "tr_abs = gradient_descent(mean_abs_c, 10., eta=1.0, n_steps=25)\n", "tr_sq = gradient_descent(mean_sq_c, 10., eta=0.005, n_steps=25)\n", "print(\"Спуск по среднему промаху, последние шаги: %s\" % np.round(tr_abs[-6:], 1))\n", "print(\"Спуск по квадрату промаха, последние шаги: %s\" % np.round(tr_sq[-6:], 2))\n", "\n", "cs = np.linspace(5, 50, 300)\n", "fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 5.5))\n", "for ax, f, tr, title in [(ax1, mean_abs_c, tr_abs, \"средний промах: наклон постоянный, спуск колеблется\"),\n", " (ax2, mean_sq_c, tr_sq, \"средний квадрат промаха: наклон тает у дна, спуск садится\")]:\n", " ax.plot(cs, [f(c) for c in cs], linewidth=2.5, color=\"0.4\")\n", " ax.plot(tr, [f(c) for c in tr], \"o-\", color=palette[1], linewidth=1.5, markersize=5, label=\"25 шагов из c = 10\")\n", " ax.set_xlabel(\"цена страницы c\"); ax.set_title(title, fontsize=12)\n", " ax.legend(fontsize=11)\n", "plt.tight_layout()\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": null, "id": "374c0ab1", "metadata": {}, "outputs": [], "source": [ "misses_A = np.array([2., 2., 2., 2.])\n", "misses_B = np.array([0., 0., 0., 6.])\n", "print(\" модель А модель Б\")\n", "print(\"средний промах %5.2f %5.2f → лучше Б\" % (misses_A.mean(), misses_B.mean()))\n", "print(\"средний квадрат %5.2f %5.2f → лучше А\" % ((misses_A ** 2).mean(), (misses_B ** 2).mean()))" ] }, { "cell_type": "markdown", "id": "ef11c9df", "metadata": {}, "source": [ "## 3. Обещание второе: как далеко шагать" ] }, { "cell_type": "code", "execution_count": null, "id": "40227ba0", "metadata": {}, "outputs": [], "source": [ "etas = [0.001, 0.008, 0.0155, 0.0175]\n", "titles = [\"слишком маленький\", \"в самый раз\", \"великоват\",\n", " \"слишком большой\"]\n", "grid = np.linspace(10, 60, 400)\n", "\n", "fig, axes = plt.subplots(1, 4, figsize=(14, 6), sharey=True)\n", "for ax, eta, title in zip(axes, etas, titles):\n", " tr = gradient_descent(mean_sq_c, 25., eta=eta, n_steps=12)\n", " ax.plot(grid, [mean_sq_c(c) for c in grid], linewidth=2, color=\"0.4\")\n", " ax.plot(tr, [mean_sq_c(c) for c in tr], \"o-\", color=palette[1], linewidth=1.5, markersize=5, clip_on=True)\n", " ax.scatter([tr[0]], [mean_sq_c(tr[0])], s=80, color=palette[0], zorder=5)\n", " if tr.max() > 65 or tr.min() < 5:\n", " title += \"\\nпосле 12 шагов c = %.0f, ошибка ×%.0f\" % (tr[-1], mean_sq_c(tr[-1]) / mean_sq_c(tr[0]))\n", " ax.set_title(r\"$\\eta = %g$: %s\" % (eta, title), fontsize=11)\n", " ax.set_xlabel(\"c\")\n", " ax.set_xlim(10, 60); ax.set_ylim(0, 12000)\n", "axes[0].set_ylabel(\"средний квадрат промаха\")\n", "plt.suptitle(\"12 шагов из одной и той же точки с разным размером шага (оси одинаковые)\", y=1.02)\n", "plt.tight_layout()\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "49a2ba92", "metadata": {}, "source": [ "## 4. Две настройки: градиент без формулы Тейлора" ] }, { "cell_type": "code", "execution_count": null, "id": "4a44a38f", "metadata": {}, "outputs": [], "source": [ "w0 = np.array([0., 40.]) # стартовая точка на карте\n", "\n", "slope_b = slope(lambda b: mean_sq2([b, w0[1]]), w0[0]) # крутим b, c заморожен\n", "slope_c = slope(lambda c: mean_sq2([w0[0], c]), w0[1]) # крутим c, b заморожен\n", "print(\"В точке (b = %g, c = %g):\" % tuple(w0))\n", "print(\" чуть-чуть повернуть b → ошибка меняется со скоростью %8.1f\" % slope_b)\n", "print(\" чуть-чуть повернуть c → ошибка меняется со скоростью %8.1f\" % slope_c)\n", "\n", "\n", "def gradient(f, w, eps=1e-4):\n", " \"\"\"То же самое для любого числа ручек: по одному ответу на каждую.\"\"\"\n", " g = np.zeros_like(w)\n", " for i in range(len(w)):\n", " dw = np.zeros_like(w)\n", " dw[i] = eps\n", " g[i] = (f(w + dw) - f(w - dw)) / (2 * eps)\n", " return g\n", "\n", "\n", "g0 = gradient(mean_sq2, w0)\n", "print(\"\\ngradient(mean_sq2, w0) = %s — те же два числа, собранные вместе\" % np.round(g0, 1))" ] }, { "cell_type": "code", "execution_count": null, "id": "5576f77f", "metadata": {}, "outputs": [], "source": [ "angles = np.linspace(0, 2 * np.pi, 361)\n", "step = 1e-3\n", "change = np.array([(mean_sq2(w0 + step * np.array([np.cos(a), np.sin(a)])) - mean_sq2(w0)) / step\n", " for a in angles])\n", "best, worst = angles[np.argmax(change)], angles[np.argmin(change)]\n", "grad_angle = np.arctan2(g0[1], g0[0])\n", "print(\"Самый крутой подъём — направление %.0f°, самый крутой спуск — %.0f°\" % (np.degrees(best), np.degrees(worst)))\n", "print(\"Направление стрелки из двух производных: %.1f°\" % np.degrees(grad_angle))\n", "\n", "B, C = np.meshgrid(np.linspace(-20, 70, 200), np.linspace(20, 45, 200))\n", "Z = np.array([[mean_sq2((b, c)) for b in B[0]] for c in C[:, 0]])\n", "levels = np.logspace(np.log10(Z.min() + 1), np.log10(Z.max()), 25)\n", "\n", "fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6))\n", "ax1.contour(B, C, Z, levels=levels, cmap=\"Blues_r\", linewidths=1)\n", "ch_max = np.abs(change).max()\n", "for a in np.linspace(0, 2 * np.pi, 24, endpoint=False):\n", " ch = (mean_sq2(w0 + step * np.array([np.cos(a), np.sin(a)])) - mean_sq2(w0)) / step\n", " length = abs(ch) / ch_max # длина стрелки — размер изменения\n", " ax1.annotate(\"\", xy=(w0[0] + 12 * length * np.cos(a), w0[1] + 3 * length * np.sin(a)), xytext=(w0[0], w0[1]),\n", " arrowprops=dict(arrowstyle=\"->\", color=palette[3] if ch > 0 else palette[0], lw=1.5))\n", "gn = g0 / np.linalg.norm(g0)\n", "ax1.annotate(\"\", xy=(w0[0] + 14 * gn[0], w0[1] + 3.5 * gn[1]), xytext=(w0[0], w0[1]),\n", " arrowprops=dict(arrowstyle=\"->\", color=\"black\", lw=3))\n", "ax1.scatter([w0[0]], [w0[1]], s=120, color=\"black\", zorder=5)\n", "ax1.scatter([w_star[0]], [w_star[1]], marker=\"*\", s=300, color=palette[3], zorder=5, label=\"дно\")\n", "ax1.set_xlabel(\"b — плата за заказ, ₽\"); ax1.set_ylabel(\"c — цена страницы, ₽\")\n", "ax1.set_title(\"24 пробных направления: длина — насколько изменилась ошибка,\\nкрасные — выросла, синие — упала; чёрная стрелка — из двух производных\", fontsize=11)\n", "ax1.legend(fontsize=11, loc=\"lower right\")\n", "\n", "ax2.plot(np.degrees(angles), change, linewidth=2.5, color=palette[0])\n", "ax2.axvline(np.degrees(grad_angle) % 360, color=\"black\", linestyle=\"--\", linewidth=2, label=\"направление стрелки из двух производных\")\n", "ax2.axvline((np.degrees(grad_angle) + 180) % 360, color=palette[3], linestyle=\"--\", linewidth=2, label=\"противоположное ей\")\n", "ax2.axhline(0, color=\"0.6\", linewidth=1)\n", "ax2.set_xlabel(\"направление шага, градусы\"); ax2.set_ylabel(\"изменение ошибки на единицу шага\")\n", "ax2.set_title(\"Изменение ошибки как функция от направления\", fontsize=12)\n", "ax2.legend(fontsize=10, loc=\"lower left\")\n", "plt.tight_layout()\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": null, "id": "46442a52", "metadata": {}, "outputs": [], "source": [ "fig, ax = plt.subplots(figsize=(10, 6.5))\n", "ax.contour(B, C, Z, levels=levels, cmap=\"Blues_r\", linewidths=1)\n", "bb, cc = np.meshgrid(np.linspace(-15, 65, 9), np.linspace(22, 44, 7))\n", "U, V = np.zeros_like(bb), np.zeros_like(cc)\n", "for i in range(bb.shape[0]):\n", " for j in range(bb.shape[1]):\n", " g = -gradient(mean_sq2, np.array([bb[i, j], cc[i, j]]))\n", " g = g / np.linalg.norm(g)\n", " U[i, j], V[i, j] = g[0] * 4, g[1] # оси в разном масштабе\n", "ax.quiver(bb, cc, U, V, color=palette[1], angles=\"xy\", scale_units=\"xy\", scale=1, width=0.004)\n", "ax.scatter([w_star[0]], [w_star[1]], marker=\"*\", s=300, color=palette[3], zorder=5)\n", "ax.set_xlabel(\"b — плата за заказ, ₽\"); ax.set_ylabel(\"c — цена страницы, ₽\")\n", "ax.set_title(\"В каждой точке карты — своя стрелка «вниз»; все они поперёк линий уровня\")\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "002a5f6b", "metadata": {}, "source": [ "## 5. Вопрос из прошлой лекции: по очереди или вместе?" ] }, { "cell_type": "code", "execution_count": null, "id": "dcffb4dd", "metadata": {}, "outputs": [], "source": [ "def descent_together(f, w0, eta, n_steps):\n", " ws = [np.array(w0, dtype=float)]\n", " for _ in range(n_steps):\n", " ws.append(ws[-1] - eta * gradient(f, ws[-1]))\n", " return np.array(ws)\n", "\n", "\n", "def best_value_of_one_knob(f, w, i, grid_width=200., n_grid=4001):\n", " \"\"\"Лучшее значение ручки i при замороженных остальных: одномерная задача, решаем перебором.\"\"\"\n", " candidates = np.linspace(w[i] - grid_width, w[i] + grid_width, n_grid)\n", " errors = [f(np.r_[w[:i], v, w[i + 1:]]) for v in candidates]\n", " w_new = w.copy()\n", " w_new[i] = candidates[int(np.argmin(errors))]\n", " return w_new\n", "\n", "\n", "def descent_one_at_a_time(f, w0, n_rounds):\n", " ws = [np.array(w0, dtype=float)]\n", " for k in range(n_rounds):\n", " ws.append(best_value_of_one_knob(f, ws[-1], 0)) # лучшее b при этом c\n", " ws.append(best_value_of_one_knob(f, ws[-1], 1)) # лучшее c при этом b\n", " return np.array(ws)\n", "\n", "\n", "n_rounds = 20\n", "tr_t = descent_together(mean_sq2, [0., 40.], eta=0.0155, n_steps=400)\n", "tr_o = descent_one_at_a_time(mean_sq2, [0., 40.], n_rounds)\n", "print(\"Спуск по стрелке, 400 шагов: ошибка %.1f, вычислений ошибки %6d\" % (mean_sq2(tr_t[-1]), 400 * 4))\n", "print(\"По очереди, %d кругов (40 подборов): ошибка %.1f, вычислений ошибки %6d\" % (n_rounds, mean_sq2(tr_o[-1]), 40 * 4001))\n", "print(\"Дно: ошибка %.1f\" % mean_sq2(w_star))\n", "\n", "fig, ax = plt.subplots(figsize=(10, 6.5))\n", "ax.contour(B, C, Z, levels=levels, cmap=\"Blues_r\", linewidths=1)\n", "ax.plot(tr_t[:, 0], tr_t[:, 1], \"-\", color=palette[1], linewidth=2, label=\"обе ручки сразу (по стрелке), 400 шагов\")\n", "ax.plot(tr_o[:, 0], tr_o[:, 1], \"o-\", color=palette[2], linewidth=2, markersize=4,\n", " label=\"по очереди: лучшее b, потом лучшее c, ... (%d кругов)\" % n_rounds)\n", "ax.scatter([0], [40], s=100, color=palette[0], zorder=5, label=\"старт\")\n", "ax.scatter([w_star[0]], [w_star[1]], marker=\"*\", s=300, color=palette[3], zorder=5, label=\"дно\")\n", "ax.set_xlabel(\"b — плата за заказ, ₽\"); ax.set_ylabel(\"c — цена страницы, ₽\")\n", "ax.set_title(\"Два способа добраться до дна\")\n", "ax.legend(fontsize=11, loc=\"upper right\")\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "f0370d74", "metadata": {}, "source": [ "## 6. Спуск по карте: узкая долина и масштаб" ] }, { "cell_type": "code", "execution_count": null, "id": "4e702a1e", "metadata": {}, "outputs": [], "source": [ "fig, axes = plt.subplots(1, 2, figsize=(15, 6))\n", "for ax, eta, n in zip(axes, [0.003, 0.016], [200, 200]):\n", " tr = descent_together(mean_sq2, [0., 40.], eta, n)\n", " ax.contour(B, C, Z, levels=levels, cmap=\"Blues_r\", linewidths=1)\n", " ax.plot(tr[:, 0], tr[:, 1], \"o-\", color=palette[1], linewidth=1, markersize=2,\n", " label=r\"спуск, $\\eta = %g$, %d шагов\" % (eta, n))\n", " ax.scatter([tr[0, 0]], [tr[0, 1]], s=100, color=palette[0], zorder=5, label=\"старт\")\n", " ax.scatter([w_star[0]], [w_star[1]], marker=\"*\", s=300, color=palette[3], zorder=5, label=\"дно\")\n", " ax.set_xlabel(\"b — плата за заказ, ₽\"); ax.set_ylabel(\"c — цена страницы, ₽\")\n", " ax.set_title(\"ошибка %.1f после %d шагов\" % (mean_sq2(tr[-1]), n))\n", " ax.legend(fontsize=10, loc=\"upper right\")\n", "plt.suptitle(\"Ландшафт ошибки для двух настроек — узкая долина\", y=1.01)\n", "plt.tight_layout()\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": null, "id": "05c2798a", "metadata": {}, "outputs": [], "source": [ "x_mean, x_std = pages.mean(), pages.std()\n", "z_pages = (pages - x_mean) / x_std\n", "\n", "\n", "def mean_sq2_scaled(w):\n", " return mean_sq2(w, x=z_pages)\n", "\n", "\n", "Xz = np.vstack([np.ones_like(z_pages), z_pages]).T\n", "wz_star = np.linalg.solve(Xz.T @ Xz, Xz.T @ paid)\n", "\n", "tr_raw = descent_together(mean_sq2, [0., 0.], eta=0.015, n_steps=3000)\n", "tr_scaled = descent_together(mean_sq2_scaled, [0., 0.], eta=0.3, n_steps=3000)\n", "print(\"Сырое число страниц, 100 шагов: b = %.2f, c = %.2f\" % tuple(tr_raw[100]))\n", "print(\"Сырое число страниц, 3000 шагов: b = %.2f, c = %.2f\" % tuple(tr_raw[-1]))\n", "b_s, c_s = tr_scaled[100]\n", "print(\"Отмасштабированный, 100 шагов: b = %.2f, c = %.2f (пересчитано обратно в страницы)\"\n", " % (b_s - c_s * x_mean / x_std, c_s / x_std))\n", "\n", "Bz, Cz = np.meshgrid(np.linspace(-50, 450, 200), np.linspace(-150, 350, 200))\n", "Zz = np.array([[mean_sq2_scaled((b, c)) for b in Bz[0]] for c in Cz[:, 0]])\n", "levels_z = np.logspace(np.log10(Zz.min() + 1), np.log10(Zz.max()), 25)\n", "\n", "fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 6))\n", "ax1.contour(Bz, Cz, Zz, levels=levels_z, cmap=\"Blues_r\", linewidths=1)\n", "ax1.plot(tr_scaled[:30, 0], tr_scaled[:30, 1], \"o-\", color=palette[1], linewidth=1.5, markersize=4, label=r\"спуск, $\\eta = 0.3$, 30 шагов\")\n", "ax1.scatter([wz_star[0]], [wz_star[1]], marker=\"*\", s=300, color=palette[3], zorder=5, label=\"дно\")\n", "ax1.set_xlabel(\"настройка при единице\"); ax1.set_ylabel(\"настройка при z\")\n", "ax1.set_title(\"После масштабирования долина круглая\")\n", "ax1.legend(fontsize=10)\n", "\n", "steps = np.arange(1, 3001)\n", "floor = mean_sq2(w_star)\n", "ax2.plot(steps, [mean_sq2(w) - floor + 1e-3 for w in tr_raw[1:]], linewidth=2.5, color=palette[0],\n", " label=r\"сырое число страниц, $\\eta = 0.015$ (больше нельзя — разлетится)\")\n", "ax2.plot(steps, [mean_sq2_scaled(w) - floor + 1e-3 for w in tr_scaled[1:]], linewidth=2.5, color=palette[2],\n", " label=r\"отмасштабированный вход, $\\eta = 0.3$\")\n", "ax2.set_xscale(\"log\"); ax2.set_yscale(\"log\")\n", "ax2.set_xlabel(\"номер шага (лог. шкала)\"); ax2.set_ylabel(\"ошибка минус её минимум (лог. шкала)\")\n", "ax2.set_title(\"Одни и те же чеки, одна процедура, разный масштаб входа\")\n", "ax2.legend(fontsize=10)\n", "plt.tight_layout()\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "89d6d816", "metadata": {}, "source": [ "## 7. Больше настроек — та же функция: перелом тарифа" ] }, { "cell_type": "code", "execution_count": null, "id": "6db79450", "metadata": {}, "outputs": [], "source": [ "pages_d = np.arange(1, 13, dtype=float)\n", "paid_d = 30 * pages_d - 20 * np.maximum(0, pages_d - 3) + rng.normal(0, 4, len(pages_d))\n", "\n", "Xd = np.vstack([np.ones_like(pages_d), pages_d]).T\n", "line_d = np.linalg.solve(Xd.T @ Xd, Xd.T @ paid_d) # лучшая прямая по формуле\n", "print(\"Лучшая прямая: %.1f + %.2f·x, средний квадрат промаха %.1f ₽²\" % (line_d[0], line_d[1], mean_sq2(line_d, pages_d, paid_d)))\n", "\n", "xs_d = np.linspace(0.5, 12.5, 300)\n", "fig, ax = plt.subplots(figsize=figsize)\n", "ax.scatter(pages_d, paid_d, marker=\"*\", s=200, color=\"black\", zorder=5, label=\"чеки четвёртого копицентра\")\n", "ax.plot(xs_d, predict2(line_d, xs_d), linewidth=2.5, color=palette[0], label=\"лучшая прямая\")\n", "ax.set_xlabel(\"число страниц\"); ax.set_ylabel(\"стоимость, ₽\")\n", "ax.set_title(\"Прямая не справляется: до трёх страниц занижает, после — завышает, потом снова занижает\")\n", "ax.legend(fontsize=11, loc=\"upper left\")\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": null, "id": "3d6c8538", "metadata": {}, "outputs": [], "source": [ "fig, axes = plt.subplots(1, 3, figsize=(14, 6))\n", "axes[0].plot(xs_d, np.maximum(0, xs_d - 3), linewidth=2.5, color=palette[1])\n", "axes[0].set_title(\"блок max(0, x − 3): ноль, потом включается\", fontsize=12)\n", "axes[1].plot(xs_d, -20 * np.maximum(0, xs_d - 3), linewidth=2.5, color=palette[1])\n", "axes[1].set_title(\"−20 · max(0, x − 3): блок с весом\", fontsize=12)\n", "axes[2].plot(xs_d, 30 * xs_d, linewidth=1.5, color=palette[0], linestyle=\"--\", label=\"30·x\")\n", "axes[2].plot(xs_d, -20 * np.maximum(0, xs_d - 3), linewidth=1.5, color=palette[1], linestyle=\"--\", label=\"−20·max(0, x − 3)\")\n", "axes[2].plot(xs_d, 30 * xs_d - 20 * np.maximum(0, xs_d - 3), linewidth=3, color=palette[3], label=\"сумма\")\n", "axes[2].scatter(pages_d, paid_d, marker=\"*\", s=120, color=\"black\", zorder=5)\n", "axes[2].set_title(\"прямая + блок = ломаная с переломом\", fontsize=12)\n", "axes[2].legend(fontsize=10, loc=\"upper left\")\n", "for ax in axes:\n", " ax.set_xlabel(\"число страниц\"); ax.axvline(3, color=\"0.7\", linestyle=\":\")\n", "plt.tight_layout()\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": null, "id": "9af4ec19", "metadata": {}, "outputs": [], "source": [ "def predict_kink(p, x):\n", " return p[0] * x + p[1] * np.maximum(0, x - p[2])\n", "\n", "\n", "def mean_sq_kink(p):\n", " return np.mean((paid_d - predict_kink(p, pages_d)) ** 2)\n", "\n", "\n", "p = np.array([10., 0., 6.]) # старт: «10 ₽ за страницу, перелома нет, где-то на 6»\n", "snapshots_kink = {0: p.copy()}\n", "for step in range(1, 5001):\n", " p = p - 1e-3 * gradient(mean_sq_kink, p)\n", " if step in (100, 1000, 5000):\n", " snapshots_kink[step] = p.copy()\n", "\n", "for step, q in snapshots_kink.items():\n", " print(\"шаг %4d: c = %5.2f, d = %6.2f, перелом t = %4.2f, ошибка %7.1f ₽²\" % (step, q[0], q[1], q[2], mean_sq_kink(q)))\n", "print(\"\\nПравда: c = 30, d = −20, t = 3\")\n", "\n", "fig, ax = plt.subplots(figsize=figsize)\n", "ax.scatter(pages_d, paid_d, marker=\"*\", s=200, color=\"black\", zorder=5, label=\"чеки\")\n", "for i, (step, q) in enumerate(snapshots_kink.items()):\n", " ax.plot(xs_d, predict_kink(q, xs_d), linewidth=2.5, color=palette[i], label=\"после %d шагов\" % step)\n", "ax.set_xlabel(\"число страниц\"); ax.set_ylabel(\"стоимость, ₽\")\n", "ax.set_title(\"Три ручки, одна из них — положение перелома; спуск находит его сам\")\n", "ax.legend(fontsize=11, loc=\"upper left\")\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "31855174", "metadata": {}, "source": [ "### Переломов несколько — блоков несколько" ] }, { "cell_type": "code", "execution_count": null, "id": "64ee0e67", "metadata": {}, "outputs": [], "source": [ "fig, ax = plt.subplots(figsize=(11, 6))\n", "ax.set_xlim(-0.5, 3.5); ax.set_ylim(-0.7, 8.7); ax.axis(\"off\")\n", "ys_hidden = np.arange(8)\n", "ax.scatter([0], [3.5], s=900, color=palette[0], zorder=5); ax.text(0, 3.5, \"x\", ha=\"center\", va=\"center\", fontsize=14, color=\"white\")\n", "ax.scatter(np.ones(8), ys_hidden, s=900, color=palette[1], zorder=5)\n", "for j in ys_hidden:\n", " ax.text(1, j, \"max(0,·)\", ha=\"center\", va=\"center\", fontsize=8, color=\"white\")\n", " ax.annotate(\"\", xy=(0.88, j), xytext=(0.12, 3.5), arrowprops=dict(arrowstyle=\"->\", color=\"0.5\", lw=1))\n", " ax.text(0.84, j + 0.3, \"w%d·x + b%d\" % (j + 1, j + 1), fontsize=8, color=\"0.3\", ha=\"right\")\n", " ax.annotate(\"\", xy=(1.88, 3.5), xytext=(1.12, j), arrowprops=dict(arrowstyle=\"->\", color=\"0.5\", lw=1))\n", " ax.text(1.16, j + 0.3, \"× v%d\" % (j + 1), fontsize=8, color=\"0.3\", ha=\"left\")\n", "ax.scatter([2], [3.5], s=900, color=palette[3], zorder=5); ax.text(2, 3.5, \"Σ + b\", ha=\"center\", va=\"center\", fontsize=11, color=\"white\")\n", "ax.annotate(\"\", xy=(2.88, 3.5), xytext=(2.12, 3.5), arrowprops=dict(arrowstyle=\"->\", color=\"0.5\", lw=1.5))\n", "ax.text(3.0, 3.5, \"стоимость\", ha=\"left\", va=\"center\", fontsize=13)\n", "ax.text(0, 8.3, \"вход\", ha=\"center\", fontsize=12); ax.text(1, 8.3, \"восемь блоков — «нейронов»\", ha=\"center\", fontsize=12); ax.text(2, 8.3, \"сумма\", ha=\"center\", fontsize=12)\n", "ax.set_title(\"Это и есть нейросеть: один вход, восемь нейронов с ReLU, один линейный выход\", fontsize=13)\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": null, "id": "7529468c", "metadata": {}, "outputs": [], "source": [ "def tariff(x):\n", " return 30 * np.minimum(x, 3) + 20 * np.clip(x - 3, 0, 7) + 10 * np.maximum(x - 10, 0)\n", "\n", "\n", "pages_e = np.arange(1, 21, dtype=float)\n", "paid_e = tariff(pages_e) + rng.normal(0, 5, len(pages_e))\n", "\n", "x_m, x_s, y_m, y_s = pages_e.mean(), pages_e.std(), paid_e.mean(), paid_e.std() # урок раздела 6\n", "z_e, u_e = (pages_e - x_m) / x_s, (paid_e - y_m) / y_s\n", "\n", "H = 14\n", "relu = lambda z: np.maximum(0, z)\n", "\n", "\n", "def net(p, z):\n", " W1, b1, W2, b2 = p[:H], p[H:2 * H], p[2 * H:3 * H], p[3 * H]\n", " return relu(np.outer(z, W1) + b1) @ W2 + b2\n", "\n", "\n", "def predict_net(p, x): # в рублях и страницах\n", " return net(p, (x - x_m) / x_s) * y_s + y_m\n", "\n", "\n", "def mean_sq_net(p):\n", " return np.mean((u_e - net(p, z_e)) ** 2)\n", "\n", "\n", "p_net = rng.normal(0, .5, 3 * H + 1)\n", "eta, snapshots = 0.2, {}\n", "for step in range(12001):\n", " if step in (0, 50, 500, 12000):\n", " snapshots[step] = p_net.copy()\n", " p_net = p_net - eta * gradient(mean_sq_net, p_net)\n", "\n", "for step, q in snapshots.items():\n", " print(\"шаг %4d: средний квадрат промаха %7.1f ₽²\" % (step, mean_sq_net(q) * y_s ** 2))" ] }, { "cell_type": "code", "execution_count": null, "id": "012854b1", "metadata": {}, "outputs": [], "source": [ "xs = np.linspace(0.5, 20.5, 400)\n", "zs = (xs - x_m) / x_s\n", "W1, b1, W2, b2 = p_net[:H], p_net[H:2 * H], p_net[2 * H:3 * H], p_net[3 * H]\n", "kinks = x_m - b1 / W1 * x_s # где включается каждый блок, в страницах\n", "\n", "fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(15, 5.5))\n", "ax1.scatter(pages_e, paid_e, marker=\"*\", s=120, color=\"black\", zorder=5, label=\"чеки\")\n", "ax1.plot(xs, tariff(xs), linewidth=1.5, color=\"0.5\", linestyle=\"--\", label=\"настоящий тариф\")\n", "for i, (step, q) in enumerate(snapshots.items()):\n", " ax1.plot(xs, predict_net(q, xs), linewidth=2.5, color=palette[i], label=\"после %d шагов\" % step)\n", "ax1.set_ylim(0, 350)\n", "ax1.set_xlabel(\"число страниц\"); ax1.set_ylabel(\"стоимость, ₽\")\n", "ax1.set_title(\"Что выдаёт сеть по ходу обучения\")\n", "ax1.legend(fontsize=10, loc=\"upper left\")\n", "\n", "for j in range(H):\n", " ax2.plot(xs, W2[j] * relu(W1[j] * zs + b1[j]) * y_s, color=\"0.4\", linewidth=1,\n", " label=\"один блок v·max(0, w·x + b)\" if j == 0 else None)\n", "ax2.axhline(b2 * y_s + y_m, color=\"0.6\", linewidth=1, label=\"свободный член\")\n", "ax2.plot(xs, predict_net(p_net, xs), linewidth=2.5, color=palette[1], label=\"сумма: предсказание сети\")\n", "ax2.scatter(pages_e, paid_e, marker=\"*\", s=120, color=\"black\", zorder=5)\n", "ax2.set_xlabel(\"число страниц\"); ax2.set_ylabel(\"вклад в стоимость, ₽\")\n", "ax2.set_title(\"Из чего складывается предсказание: восемь блоков, у каждого свой перелом\")\n", "ax2.legend(fontsize=10, loc=\"upper left\")\n", "plt.tight_layout()\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": null, "id": "0a5fd9d1", "metadata": {}, "outputs": [], "source": [ "print(\"Перелом каждого из восьми блоков (в страницах), по какую сторону он включён и сколько добавляет к цене страницы:\")\n", "for j in np.argsort(kinks):\n", " inside = 0.5 <= kinks[j] <= 20.5\n", " print(\" блок %d: перелом на %6.1f стр., включён %s, добавляет %6.2f ₽/стр. %s\"\n", " % (j + 1, kinks[j], \"правее\" if W1[j] > 0 else \"левее \", W2[j] * W1[j] * y_s / x_s,\n", " \"\" if inside else \"(вне диапазона чеков)\"))\n", "\n", "fig, ax = plt.subplots(figsize=figsize)\n", "ax.scatter(pages_e, paid_e, marker=\"*\", s=120, color=\"black\", zorder=5, label=\"чеки\")\n", "ax.plot(xs, predict_net(p_net, xs), linewidth=2.5, color=palette[1], label=\"предсказание сети\")\n", "for j in range(H):\n", " if 0.5 <= kinks[j] <= 20.5:\n", " ax.axvline(kinks[j], color=palette[1], linestyle=\":\", linewidth=1.5,\n", " label=\"переломы, найденные сетью\" if j == np.argsort(kinks)[0] else None)\n", "for t_true in (3, 10):\n", " ax.axvline(t_true, color=palette[3], linestyle=\"--\", linewidth=2, label=\"настоящие переломы тарифа\" if t_true == 3 else None)\n", "ax.set_ylim(0, 350)\n", "ax.set_xlabel(\"число страниц\"); ax.set_ylabel(\"стоимость, ₽\")\n", "ax.set_title(\"Переломы, которые сеть расставила сама, против настоящих переломов тарифа\")\n", "ax.legend(fontsize=10, loc=\"upper left\")\n", "plt.show()" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.13.5" } }, "nbformat": 4, "nbformat_minor": 5 }